Nuprl Lemma : rpoly-deriv_wf
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀x:ℝ. (rpoly-deriv(n;a;x) ∈ ℝ)
Proof
Definitions occuring in Statement :
rpoly-deriv: rpoly-deriv(n;a;x),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rpoly-deriv: rpoly-deriv(n;a;x),
uall: ∀[x:A]. B[x],
nat: ℕ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
prop: ℙ,
rev_implies: P ⇐ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
subtract: n - m,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
int-to-real_wf,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
rpolynomial_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
le_wf,
poly-deriv_wf,
subtract-add-cancel,
subtype_base_sq,
int_subtype_base,
add-commutes,
add-associates,
add-swap,
equal_wf,
real_wf,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
natural_numberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
intEquality,
hypothesisEquality,
independent_functionElimination,
productElimination,
independent_isectElimination,
independent_pairFormation,
impliesFunctionality,
dependent_set_memberEquality,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addEquality,
instantiate,
cumulativity,
functionEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}x:\mBbbR{}. (rpoly-deriv(n;a;x) \mmember{} \mBbbR{})
Date html generated:
2017_10_03-PM-00_14_54
Last ObjectModification:
2017_07_28-AM-08_37_24
Theory : reals
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